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Hitcghiker x27 s guide Bull Sci math ?? www elsevier com locate bulsci Hitchhiker ? s guide to the fractional Sobolev spaces ?? Eleonora Di Nezza a ? Giampiero Palatucci a b Enrico Valdinoci a c a Dipartimento di Matematica Università di Roma ??Tor Vergat

Bull Sci math ?? www elsevier com locate bulsci Hitchhiker ? s guide to the fractional Sobolev spaces ?? Eleonora Di Nezza a ? Giampiero Palatucci a b Enrico Valdinoci a c a Dipartimento di Matematica Università di Roma ??Tor Vergata ? ?? Via della Ricerca Scienti ?ca Roma Italy b Dipartimento di Matematica Università degli Studi di Parma Campus ?? Viale delle Scienze A Parma Italy c Dipartimento di Matematica Università degli Studi di Milano ?? Via Saldini Milano Italy Received June Available online December Abstract This paper deals with the fractional Sobolev spaces W s p We analyze the relations among some of their possible de ?nitions and their role in the trace theory We prove continuous and compact embeddings investigating the problem of the extension domains and other regularity results Most of the results we present here are probably well known to the experts but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces We also present some counterexamples in non-Lipschitz domains ? Elsevier Masson SAS All rights reserved MSC primary E secondary S S Keywords Fractional Sobolev spaces Gagliardo norm Fractional Laplacian Nonlocal energy Sobolev embeddings Contents Introduction The fractional Sobolev space W s p ?? Part of this paper has been used in a few lectures held at the University of Rome ??Tor Vergata ? Corresponding author E-mail addresses dinezza mat uniroma it E Di Nezza giampiero palatucci unimes fr G Palatucci enrico math utexas edu E Valdinoci The author has been supported by Istituto Nazionale di Alta Matematica ??F Severi ? Indam and by ERC grant ?? Vectorial problems ? The author has been supported by the ERC grant ?? Elliptic Pde ? s and Symmetry of Interfaces and Layers for Odd Nonlinearities ? and the FIRB project ??A B Analysis and Beyond ? - ?? see front matter ? Elsevier Masson SAS All rights reserved doi j bulsci C E Di Nezza et al Bull Sci math ?? The space H s and the fractional Laplacian operator Asymptotics of the constant C n s Extending a W s p ? function to the whole of Rn Fractional Sobolev inequalities Compact embeddings H? lder regularity Some counterexamples in non-Lipschitz domains References Introduction These pages are for students and young researchers of all ages who may like to hitchhike their way from to s ?? To wit for anybody who only endowed with some basic undergraduate analysis course and knowing where his towel is would like to pick up some quick crash and essentially self-contained information on the fractional Sobolev spaces W s p The reasons for such a hitchhiker to start this adventurous trip might be of di ?erent kind s he could be driven by mathematical curiosity or could be tempted by the many applications that fractional calculus seems to have recently experienced In a sense fractional Sobolev spaces have been a classical topic in functional and